Your move, brainlet

Your move, brainlet

Other urls found in this thread:

en.wikipedia.org/wiki/Pseudometric_space
wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=i, b=1, c=0
wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=i, b=0, c=0
wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=1, b=0, c=0
wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=2i, b=0, c=0
wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=2, b=0, c=0
twitter.com/NSFWRedditImage

Euclid BTFO

Blue line would be length square root 2

i cant think of something better you already won

The complex numbers can't be properly ordered which means they can't be used in the length of a segment.

Oh, yes. The beautiful theorem, [math] -1 + 1 = 2 [/math]. I believe it was proven by Euler on a rainy day.

he meant inverse.
and I believe that it was, in fact, Euler that proved 1-1+1-1...=1/2

>
you can't have a sidelength of a triangle be negative so the problem is invalid :^)

>muh applied mathematics
fuck off brainlet

>Euler that proved 1-1+1-1...=1/2
Euler is great but he genuinely pissed me off because he did not really prove those things. He symbolically made them happen. None of his calculations for that, or the other crazy shit he did, were rigorous. And if you have ever tried to write nonsense in math you know that sometimes that nonsense somehow reaches the correct answer, and sometimes you just get nonsense back. And that motherfucker got so lucky with this shit. Like 9 out of 10 times Euler wrote nonsense on a paper it turned out to be correct but he was walking a fine line. I remember I once read a little about the life of the guy who is the namesake of the Wronskian and it said that often what he published turned to be completely retarded because none of his calculations were rigorous and then people just confirmed he was batshit insane by trying to confirm his findings. He was expelled from the scientific world for publishing so much bullshit and died poor and rotting.

That guy and Euler are the same guy. Only Euler got really fucking lucky and the Wronskian guy did not. By the way, the reason he gets to be the name of the Wronskian was completely out of pity. He did not call the Wronskian, he did not even really study. He just casually used the same computation in the 2x2 case. And because that was his single thing that was not so shit, out of pity they named it after him. Poor guy.

But imagine if Euler's original proof for the sum of the reciprocals of the squares turned out to be wrong, and some other mathematician found that the sum actually equals something else. Then Euler would have been laughed at like a fucking retard and expelled from the scientific community like the Wronskian guy.

Euler is a lucky bastard.

the sides of a triangle can only be real positive numbers...

>None of his calculations for that, or the other crazy shit he did, were rigorous
But they were. All he did was define summation differently. Obviously that gives you a different system, but both can perfectly consistent and there's no reason why you shouldn't change definitions like he did (as long as the resulting system is consistent).
>Like 9 out of 10 times Euler wrote nonsense on a paper it turned out to be correct
You know the odds of that right? Even if he wasn't completely right about his statements, he still had something special about him that allowed him to be right 9 out of 10 times. I mean, Ramanujan was basically a more extreme example of that. He's still considered, rightly I think, one of the greatest mathematicians.

to be fair, basically everyone at that time did shit like that. Euler just had gread intuition and was an absolute madman, that calculated shit like [math] 2^{2^5}+1 = 641\cdot 6700417 [/math] just to btfo fermat.

It's true that much of Euler's work lacked real rigor but in fairness to him this was mostly because the mathematics that would be required to do it formally simply did not exist yet. Just look at how fucking often Euler's name appears in modern mathematics, you can hardly call him being correct so often a coincidence. He had a keen mathematical intuition for what would be formally correct even if he did not himself develop the theories completely rigorously (which generally didn't happen until hundreds of years after him)

the complex plane, there is a triangle with points at the origin, 0 + i, and 1 +0i . theperpendicular sides of the triangle therefore have lengths : i - 0i = i and 1-0 = 1

so by pathagoras' theorem, A^2 + B^2 = c^2 and i^2 + 1^2 = -1 + 1 = 0.

But because the complex plain is a metric space, the distance between 2 points can only be 0 if they are the same point.
0 + i and 1 +0i are not the same point, therefore there is a contradiction , proving once again that "complex" or "imaginary" numbers are a fallacious , self-contradictory construction

FACT: any real world application of complex numbers can be accomplished with infinite series and trigonometric functions.

EBBIN MY FRIEND JAJAJA TOTS U TROLLOLOLED US. U WIN INTERNET YOU SIR OF CLASS XDDDD

>All he did was define summation differently.
No, he didn't. He did not invent or use the current definitions of analysis. He preceded Cauchy which means that he was basically analytically retarded, as was everyone else. He just got really lucky.

>to be fair, basically everyone at that time did shit like that
Exactly. I just find interesting how Euler is fondly remembered as a genius of analysis and appears everywhere in books of analysis as the great genius who did everything. When if you look at the history books he died when Cauchy was like 7.

I particularly like his use of infinite products in computing like the factorial of 1/2. What a fucking lucky bastard.

> because the mathematics that would be required to do it formally simply did not exist yet.
Exactly. And using theorems that do not exist is sketchy at best. But yeah, this was a case of Euler not knowing that he did not know the theorems needed to do his stuff. Cauchy even called this the "generality of algebra".

>you can hardly call him being correct so often a coincidence.
Yes, I can. And they were. And I believe he was correct so often because he did so much math all the time. He did so much work that he was bound to find at least a couple of true statements.

You know, I bet that when Euler published his proof of the sum of the reciprocal of the naturals he just published the argument. But behind everyone's back you better believe that bastard computed that sum up to like 20 digits to be sure that he was not going to get the boot.

brainlet

>the distance between 2 points can only be 0 if they are the same point.
You are like a little baby

the blue line should be -1 + i, because that is the complex number representing the vector that overlaps the blue line.

There is no i length

in a metric space that is true and C is a metric space.

>in a metric space that is true and C is a metric space.

not that guy but the picture OP gives is clearly not on the complex plane, but some space which allows for complex distances which is very much exotic enough to lead to bizarre behavior like in OP's pic.

"distance" means metric distance whenever you use it in math. OP's pic is a meme

Oh yeah? Explain this:
[math]\sum_{n=1}^\infty n = -\frac{1}{12}[/math]

>tfw cannot into magnitudes

You have to take the absolute values (magnitude) for pythagoras theorem.

sqrt( |-1i|^2 + 1^2 ) = sqrt ( 1^2 + 1^2 ) = sqrt(2)

Wouldnt i be coming out of plane? It isnt a flat polygon then. How would angles work?

Actually, Can i be x and 1 become 2 and can i just add a letter to the 0, like,idk, y so it becomes 0y and solve for cheesecake times sigma delta equals my foot up your ass squared across your moms ugly face to the power of cylinder?


Is there a fucking question? It's a fucking triangle.

Can you actually interpret what you see? It's a triangle. Now you can go outside and do some sports, eat some food so you don't look as pasty as you do, maybe try to make a few friends or something, now that you've solved for triangle successfully.

Spell it with me one moretime:

T r i a n g l e

>he doesn’t understand metrics
Laughingwhores.jpeg

You are amazingly retarded.

Assuming you used the Pythagorean Theorem to calculate the hypotenuse, you are implying a metric, basically Euclidean distance.

Go back to precalc faggot.

Stop pretending it's not a triangle.

Fake Math. Sad. This series was invented by the Chinese for the Chinese to rip us off in trade.

It’s not. In the Euclidean plane the triangle inequality holds and this violates that. You can’t use Pythagoras and not use Euclidean plane.

I’m sorry you’re still a high school math student, but this is obvious to anyone who has taken anything after babby's first geo.

I'm sorry, but it really just is a triangle with some meaningless letters and numbers thrown in the mix to give the impression of a question or in this case apparently a solution.

How do i apply the teachings of a long dead greek philosopher to access this mysterious euclidean plane where the secrets of the triangle are stored?

....___
i=\/0-1

F*k(20-k+b)*pizza:2

>0, 1, i
>negative

holy fuck, i just learnt about Wronskian for the first time earlier today, and I'm already finding it mentioned online in multiple places. this shit happens so often

if the hypotenuse is 0, it's just a line.
i = 1.

What do you mean by properly ordered?
And every set, including set of complex numbers can be well ordered

en.wikipedia.org/wiki/Pseudometric_space

Yes they can, they can even be ordered in a way that is compatible with the ordering of the reals:
Given [math]z,w\in\mathbb{C}[/math], if [math]\operatorname{Re} z \not = \operatorname{Re} w[/math], the ordering of [math]z, w[/math] is that of their real components; if [math]\operatorname{Re} z = \operatorname{Re} w[/math], their ordering is that of their imaginary components.
This is a total order, and if restricted to the reals, is equivalent to the regular real ordering.
It just happens that ordering [math]\mathbb{C}[/math] generally isn't very useful, but it is possible.

one almost always uses the metric identification when using the distance in a meaningful way, like when you work in Lp spaces

If you do that, then the hypotenuse doesn't equal zero.

This is not hard to understand and OP is a retarded faggot and people who are being contrarian shitstains trying to defend him are even worse.

I really hope this is bait.

You could start by understanding geometry at more depth than high school and understanding what a metric is.

>Oh, yes. The beautiful theorem, −1+1=2. I believe it was proven by Euler on a rainy day.
Actually, this is called the Oiler-GauB identity.

You literally cannot have distance in any meaningful sense without the triangle inequality.

It's part of the definition of metric space.

>using i as a unit of length
You have to make it a magnitude you fuck

>that much of Euler's work lacked real rigor but in fairness
But when I use my intuition, they call me a brainlet, even when I come to conclusions that rigorous approaches agree with, but take longer. This is true even when my intuition is faster and consistently true. What's up with that? How come no one gave Euclid shit for not being a rigor addict in his day, and still don't?

Just use 1i and you get the same problem you chucklefuck

>1i=1

This is what shitposting looks like

>You can’t use Pythagoras and not use Euclidean plane.
>Pythagoras
>Born c.570 BC
>Died c. 495 BC
>Euclid
>Born Mid-4th century BC
>Died Mid-3rd century BC
Yeah, I'm sure you need an EUCLIDIAN PLANE to use something that was invented before EUCLID was even born

Gr8 b8 m8

blue^2 = green * conj(red)

:^)

actually I just quoted the wrong post
I meant to reply to this guy
to say that i had the magnitude of 1

see dumbass

>the vector from -1 to 0 has length -1 - 0 = -1
Maximum brainlet

brainlet

If the Complex Plane is a metric space, and you show the distance between two distinct points to be zero, you haven't proven by contradiction that all of Complex numbers are "fallacious" or "self-contradictory", all you've show is that you've used the wrong Metric by assuming that distance in the Complex Plane is calculated with Pythagoras'. It isn't. You might think you're trolling or something. You aren't. You're just retarded.

>implying that there can be a leg of the triangle equal to i

Sad!

>implying pythagaros didn't use ancient atlantean technology to travel to the future

Ok, listen. To define a distance on some set Y we define a metric on Y as a function
d: Y x Y ---> IR+
(x,y) |--> d(x,y)
For this function to be a metric it has to have the following properties: (for all x,y,z in Y)
- d(x,y) >= 0
- d(x,y) = d(y,x)
- d(x,y) = 0 x=y
- d(x,z)

>*: C x C → C
>(a,b) → d(a,b) = \sqrt{a_x*_b_x - a_y*b_y, a_x*b_y + a_y*b_x}
should be
*: C x C → C
(a,b) → [math]a*b = (a_x*b_x - a_y*b_y, a_x*b_y + a_y*b_x)[/math]

Veeky Forums delivers.

I believe (put equivalently) that the modulus is a sensible metric on the complex numbers. The value of a complex number in terms of the real field it extends is not.

You just have a line and i=1, apparently.

>I believe (put equivalently) that the modulus is a sensible metric on the complex numbers.
Modulus is the same as the euclidean distance to the origin (= the euclidian norm). Dont know if you already mean that by the "(put equivalently)", but you may as well point it out directly to all the brainlets out there.

...

Since [math]zz^* = i[/math] has no solution, can't we extend the complex numbers in an analogous way to extending the reals with a solution to [math]x^2 = -1[/math]?

The new number would then by definition have a magnitude of i.

It may be possible, but I dont think we'll be able to define a metric on that space (Since a metric has to be a map to positive real numbers). Also I am not sure that this new space would have many interesting properties. You'd have to check tho.

The *length* is a sum of square magnitudes, in the direction (1,i)

Your fourth property is the triangle inequality as mentioned

nothing says that it's a right triangle
checkmate OP

a^2 + b^2 = c^2

a = 1 b= i c = 0
1^2 + i^2 = 0^2
1 + i^2 = 0
i^2 = -1
i = -1^1/2

this makes sense to me guys

>0 cannot represent length
>but i can
nice logic

>70 posts
>still not a single correct answer
jesus christ, move brainlets
in complex plane pythagorean theorem looks like this:

|a^2| + |b^2| = |c^2|

now you can delete this thread

>-1 = 1 this is what shitposting looks like

I don't think you understand what length is, length can only be a positive real number

Stop applying properties of R^2 to the complex plane

It's the norm squared not the norm of the square you dumbass. Also, that's how it look in any fucking inner product space.

not him but you're actually retarded

how do you draw imaginary numbers ?

What exactly is wrong?

Which is just measuring distance with vectors, which can only happen in a metric space which has been mentioned.

You’re going to be late for Calc I tomorrow freshman.

no sign of a right angle. nice b8 trying to activate my almonds

Just imagine a darn 2d space with y axis being imaginary axis. It's pretty clear that
|a|^2 + |b|^2 = |c|^2

>this thread is still up
>some retards are actually falling for it/giving incorrect answers
Do you really have a hard time with complex numbers algebra Veeky Forums?

It's still a real valued function.

2 dimensional objects don't have ordered value so the absolute value is still 1.

So I ran a check to laugh at OP when something doesn't sum up and here's the results:

>Area comes out [math]\frac{i}{2}[/math] with Heron's formula
wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=i, b=1, c=0
>Area comes out [math]\frac{i}{2}[/math] again with the [math]A = \frac{hb}{2}[/math] , [math]h[/math] being [math]i[/math] in this case:
[math]A = \frac{1i}{2} = \frac{i}{2}[/math]
>If you split the triangle in two equal you get a triangle with sides [math]i, 0, 0[/math] and a triangle with sides [math]1, 0, 0[/math], both of which end up with half the area of the initial triangle
[math]a = i[/math] wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=i, b=0, c=0
[math]a = 1[/math] wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=1, b=0, c=0

>If you add two of the initial triangles together you end up with:
Same triangle flipped and glued to the imaginary (red) side, extending [math]i[/math] to [math]2i[/math] while having both legs be 0
Area becomes twice the initial triangle's area as expected, [math]i[/math] - wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=2i, b=0, c=0
>The above repeated but gluing it to the real (green) side instead, creating a triangle of [math]2, 0, 0[/math]
Same outcome with same area wolframalpha.com/input/?i=solve sqrt(s((s-a)*(s-b)*(s-c))) where s=(a+b+c)/2, a=2, b=0, c=0

I know I know, muh magnitude n shit but can someone explain why the fuck does this work so well and doesn't break the standard formulas?

so what is abs(i) einstein?

>Same triangle flipped and glued to the imaginary (red) side
fuck I meant to say to the real side, like pic related and the reverse for the triangle with [math]2, 0, 0[/math]

this is the same logic you're doing OP

Nah. It was actually Eueler who proved that series diverges. Leibniz was the one who said I equaled 1/2.

the length is still 4 tho

1, brainlet

Assume that the Pythagorean theorem holds on the imaginary plane:
[math]a^2 + b^2 = c^2[/math]
[math]i^2 + 1^2 = c^2[/math]
[math]-1 + 1 = 0[/math]
But,
[math]a^2 + b^2 = c^2[/math]
[math]0^2 + 0^2 =(2i)^2[/math]
[math]0 + 0 = -4[/math]

Therefore, the Pythagorean theorem does not hold on the imaginary plane.

no, it's undefined just as dividing by zero is
|i| doesn't make any sense

Yes it does. Absolute value is distance from 0 and the distance between 0 and i is 1.

>But,
The triangle you're describing, [math]c = 2i[/math], isn't a right triangle and therefor the Pythagorean theorem doesn't apply, see:

>babby's first hyperbolic geometry

Pythagorean theorem does hold for complex plane you just need to take the absolute values of the sides. Do you even know what the theorem is or why it holds?

...

Then 1+2+3+4+5+6...=-1/12